Question: You have found the following ages (in years) of all 5 tigers at your local zoo: $ 12,\enspace 13,\enspace 11,\enspace 5,\enspace 5$ What is the average age of the tigers at your zoo? What is the standard deviation? You may round your answers to the nearest tenth.
Solution: Because we have data for all 5 tigers at the zoo, we are able to calculate the population mean $({\mu})$ and population standard deviation $({\sigma})$ To find the population mean , add up the values of all $5$ ages and divide by $5$ $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{5}} x_i}{{5}} $ $ {\mu} = \dfrac{12 + 13 + 11 + 5 + 5}{{5}} = {9.2\text{ years old}} $ Find the squared deviations from the mean for each tiger. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $12$ years $2.8$ years $7.84$ years $^2$ $13$ years $3.8$ years $14.44$ years $^2$ $11$ years $1.8$ years $3.24$ years $^2$ $5$ years $-4.2$ years $17.64$ years $^2$ $5$ years $-4.2$ years $17.64$ years $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean , we can find the variance $({\sigma^2})$ , without introducing any bias, by simply averaging the squared deviations from the mean $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{7.84} + {14.44} + {3.24} + {17.64} + {17.64}} {{5}} $ $ {\sigma^2} = \dfrac{{60.8}}{{5}} = {12.16\text{ years}^2} $ As you might guess from the notation, the population standard deviation $({\sigma})$ is found by taking the square root of the population variance $({\sigma^2})$ ${\sigma} = \sqrt{{\sigma^2}}$ $ {\sigma} = \sqrt{{12.16\text{ years}^2}} = {3.5\text{ years}} $ The average tiger at the zoo is 9.2 years old. There is a standard deviation of 3.5 years.